Finite Element Method for Huge Domains and Related Topics

1. 主要研究成果：(1)利用电荷模拟方法对圆盘外域拉普拉斯问题相关的Steklov算子进行离散化。1998年提出了求解二维外部拉普拉斯问题的FEM-CSM（有限元法-电荷模拟法）组合方法，并在整个项目期间对其数学分析进行了研究。研究了连续傅立叶系数与离散傅立叶系数之间的关系，这是研究的关键。(2)通过数值计算验证了FEM-FSM联合方法的有效性（与Masuda， S.联合）。(3)通过对势函数和流函数进行精确的有限元计算，确定机翼的保角映射。(4)提出了确定圆盘外域约简波问题相关Steklov算子的基本求解方法（FSM）。(1)利用神经网络求解偏微分方程的数值方法。边界条件拟合采用形状因子的近似函数形式。MHD平衡方程和Napier-Stokes方程的神经网络求解器。加速神经网络的求解方法采用高斯-牛顿法。(2) Helmholtz方程在区域分解计算中的迭代方法。开放边界部分与外部无限区域连通的有界区域内波传播的数值分析与计算。二阶问题的Newmark方法的稳定性和收敛性证明（由Kako， T.获得）。(3)利用线性系统理论分析龙格-库塔方法的稳定性。用龙格-库塔法求解时滞微分方程的稳定性分析（由Koto获得）。(4)求解外部亥姆霍兹问题的波动方程方法的数学证明。2 .外Helmholtz问题波动方程求解器的一种新的人工边界条件的提出和证明（小山获得）。保持uec - na研讨会（电子通信大学数值分析研讨会）。一九九八及一九九九年每个学年，一般在星期五上午举行十四次研讨会。在研讨会上，山形大学的神谷教授、名古屋大学的坂井教授、九州大学的山本教授、名古屋大学的松尾教授、九州大学的金山教授等人的演讲都得到了这笔资金的支持。除这些定期研讨会外，还举行了为期一天和两天的特别讲习班，讨论项目的主题和有关专题。少

1. Main Results Obtained in the Joint Works around the Head Investigator :(1) Discretization method through charge simulation method for Steklov operator associated with Laplace problems in exterior domain of a disc. An FEM-CSM (Finite Element Method - Charge Simulation Method) combined method for 2D exterior Laplace problems was proposed in 1998 and its mathematical analysis has been studied during whole the term of project. Relation between continuous and discrete fourier coefficients, which is a key for the study, has been investigated.(2) Confirmation of the effectiveness of the FEM-FSM combined method through numerical computation (jointly with Masuda, S.).(3) Determination of the conformal mapping of wing through precise finite element computation for the potential function and stream function.(4) Proposal of the fundamental solution method (FSM, in short) for the determination of Steklov operator associated with the reduced wave problems in exterior domain of a disc.2. Remarkabl … More e Progress Obtained in the Works by Investigators :(1) Numerical method for partial differential equations through neural network. Boundary condition fitting with the use of shape factor in the form of approximate function. Neural network solver for MHD equilibrium equation and Napier-Stokes equation. Acceleration of the neural net solution method with the use of Gauss-Newton method. (Obtained by Takeda and Fukuhara)(2) An iterative method for Helmholtz equation in domain decomposition computation. Numerical analysis and computation of the wave propagation in a bounded domain with open boundary portion connected with exterior infinite region. Proof of stability and convergence for Newmark's method for second order problem (obtained by Kako, T.).(3) Stability analysis for Runge-Kutta method through linear system theory. Stability analysis for Runge-Kutta method applied to delay-differential equations (obtained by Koto).(4) Mathematical justification for a wave equation method solving exterior Helmholtz problems. Proposal and justification of a new artificial boundary condition to the wave equation solver for exterior Helmholtz problem (obtained by Koyama).3. Keeping the UEC-NA-Seminar (University of Electro-Communications Numerical Analysis Seminar). In each academic year of 1998 and 1999, 14 seminars were held in the Friday morning, generally. Within seminars, talks given by Professor Kamitani, A. of Yamagata University, Professor Sakajo, T. of Nagoya University, Professor Yamamoto, N. of then Kyushu university, Professor Matsuo, T. of Nagoya University, Professor Kanayama, H. of Kyushu University were supported by this grant. In addition to these regular seminar, special one-day and two-day workshops were held to discuss the subject of the project and related topics. Less